Pattern-Oriented Modelling

At the end of the 1980s, when the first individual-based or agent-based models appeared, scientists were very excited about the promise of such computer simulation models to unify ecological theory. However, the initial enthusiasm quickly dissipated due to a number of problems, including a lack of methods to cope with the issues of model complexity and er-ror propagation, and the models were often designed ad hoc, not tied to theory, and lacking generality. To be fair, one could not expect full solutions to these problems within the 10 or 20 years these models would be around; analytical equation models had some 200 years to find them.

Together with my colleague Volker Grimm and others (Grimm et al. 2005), we developed pattern-oriented modelling as a strategy to find the Medawar zone for individual-based simu-lation models. We explicitly followed the basic research programme of science: the explana-tion of observed patterns. Patterns are characteristic structures that contain informaexplana-tion on the internal organisation of a system. In practice, we compare the ability of alternative models with different levels of complexity to reproduce several patterns at the same time. The focus on multiple patterns is important because it is well known that substantially different models can reproduce the same pattern. However, two or more patterns that describe different char-acteristics of the system are not that easily to reproduce at the same time by different models.

I will now illustrate this modelling strategy with an example of my own work conducted under a European Research Council (ERC) advanced grant. A big question in ecology is to explain the high species richness of tropical forests. Traditionally, ecologists assume that each species in a given ecosystem is different and occupies its own ecological niche, thereby limiting the interactions with others required for its persistence. However, around 2001 the publication of a book by tropical ecologist Stephen Hubbell called The Unified Neutral Theory of Biodiversity and Biogeography caused a big ‘scandal’ that shook the fundamentals of ecology. What upset many ecologists about this book was that Hubbell claimed that neu-tral models, a class of analytical models that assumed that all species are identical (and have no niches), could explain important properties (patterns) of species rich communities. For example, these models can predict the distributions of rare and abundant species in tropical forests or coral reefs. Clearly, neutral models were not really welcome because all the work ecologists had done for so long on species differences and niches suddenly seemed irrelevant.

Ecologists before Hubbell usually focussed on differences among species and started with the most complex situation, whereas Hubbell used the simplest case as a starting point.

He tested how far he could go with his radical assumptions to find out how much detail must be added to a neutral model to explain important properties of species in rich communities.

My strategy was to combine the strengths of different models. We used the analytical predic-tions of neutral theory as point of reference and started with a spatially explicit and individual-based version of a neutral model. This allowed us to compare the model output with many more patterns extracted from inventory maps of tropical forests (Fig. 4) that were possible with the analytical neutral theory. We used also null model approaches to identify spatial patterns in the

Thorsten Wiegand

distribution maps of trees (Wiegand and Moloney 2014). Finally, we developed alternative model versions that included the simplest neutral models and also models where species were different and tested their ability to reproduce the patterns observed in the forest inventories.

Fig. 4 Spatial inventory data of the Sinharaja tropical forest in Sri Lanka and patterns that can be extracted from such data. The size and status of every tree is measured every five years. This allows extraction of data on survival and growth of trees used to parameterise the models. Additionally, it includes the size distribution patterns of individual species, information on the spatial aggregation pattern of individual species and co-occurrence patterns of different species that live close to one another. We can also determine how many species can be found on average in an area of a given size and how the local species composition changes in space. Finally, the inventory data contain information on the ecological similarity of neighboured individuals and how strongly they compete.

One challenge with this pattern-oriented approach was to parameterise 200 or 300 species to describe the interactions of a total of 20,000 to 200,000 individuals. We were very lucky to get access to the data of the CTFS-ForestGEO network of the Centre of Tropical Forest Science (CTFS), one of the largest data enterprises in ecology. Today, that network comprises 63 field sites all over the world that all follow the exact same protocol. They comprise completely mapped inventory plots of tropical, subtropical and temperate forest of up to 50 hectares, recording and mapping every tree bigger than one centimetre in diameter, and then

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ing the development of each tree every five years. This has generated a huge amount of data.

Figure 4 shows as an example a representation of the larger trees in one forest in Sri Lanka.

We used individual-based dynamic biodiversity models with different complexities to find out how complex our model must be to recreate forests with observed spatial structures as those shown in Figure 4. Our hypothesis was that the neutral models are oversimplified and would completely fail to capture any of the complex spatial structures in species diversity.

To meet the technical challenges of the pattern comparisons, we developed new methods of stochastic inference that allowed us to use well-established optimisation tools to fit the model to the observed summary statistics. We also made no attempt to parameterise individu-al species. That would have been impossible as there were 200 species, many of them so rare that little to nothing was known about them. So we used distributions for their parameters. If the variance of the distribution was zero, we obtained a neutral model, because then this prop-erty was the same for all species. Increasing the variance yielded more and more variability in the properties of the different species.

The surprising and unexpected result was that the simple neutral model already provides a very good approximation of the complex spatial structures of species in rich tropical forests (May et al. 2015). The model was able to fit all individual patterns with very high precision and to fit several patterns together with sufficient precision (but not all). So, it looks that Hub-bell was right, after all. Our structurally realistic model failed in an especially informative way. This allowed us to test specific hypotheses on the relative importance of species differ-ences and niches in explaining additional properties (patterns) of species-rich communities.

In summary, to cope with complexity, one should try to combine the strengths of the dif-ferent types of models, employ analytical models and predictions from ecological theory as starting points, then access new data sources by using statistical and phenomenological mod-els to identify patterns in the data, and strive to explain these patterns rather than modelling a complete system. To do this, one can take advantage of newer methods of statistical inference for model selection that can actually tell us how much complexity is needed. As always, one must apply Occam’s razor: keep models as simple as possible but as complex as necessary.

References

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Grimm, V., and Railsback, S. F.: Individual-based Modeling and Ecology. Princeton, NJ: Princeton University Press 2005

Grimm, V., Revilla, E., Berger, U., Jeltsch, F., Mooij, W. M., Railsback, S. F., Thulke, H.-H., Weiner, J., Wiegand, T., and DeAngelis, D. L.: Pattern-oriented modeling of agent-based complex systems: Lessons from ecology. Science 310, 987–991 (2005)

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Hubbell, S. P.: The Unified Neutral Theory of Biodiversity and Biogeography. Princeton, NJ: Princeton University Press 2001

May, F., Huth, A., and Wiegand, T.: Moving beyond abundance distributions – neutral theory and spatial patterns in a tropical forest. Proceedings Royal Society B 282/1802, 20141657 (2015)

Wiegand, T., and Moloney, K. A.: A Handbook of Spatial Point Pattern Analysis in Ecology. Boca Raton, FL:

Chapman and Hall/CRC Press 2014